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Tinbergen Institute The Tinbergen Institute is the institute for economic research of the Erasmus Universiteit Rotterdam, Universiteit van Amsterdam and Vrije Universiteit Amsterdam.
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Most TI discussion papers can be downloaded at
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Surri ri Sursrvlwlrq 7= D qhfhvvdu| dqg vx!flhqw frqglwlrq iru wkh sulfh vfkhgxoh wr eh xqlirup lv wkh iroorzlqj= Csf +v, C +v, Csf +v, C +v, . @ . = Cv Cv Cv Cv
+47,
Iurp htxdwlrq +48,/ e| gl huhqwldwlqj sf +v,t +v, zlwk uhvshfw wr v dqg uhsodflqj wr }+t> y, @ t^y +@5,t` zh jhw Csf +v, Ct +v, f t +v, @ s +v, . y t +v, Cv Cv dqg
5
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54
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+48,
6
Ct +h v > v3 , 8 gh v = Cv
+49,
Lq wklv fdvh/ wkh htxdwlrq deryh gh qhv wkh IRF ri rxu remhfwlyh ixqfwlrq M/ vlqfh wkh Nxkq0Wxfnhu pxowlsolhuv/ b Er dqg qEr duh erwk }hur1
55
Qrz/ pxowlso|lqj erwk vlghv ri htxdwlrq +48, e| t +v, dqg uhsodflqj iurp htxdwlrqv +53, dqg +54, |lhogv wkh iroorzlqj C +v, Ct +v, f s +v, . y t +v, . t +v, @ Cv Cv 6 5 ]r Ct +h v > v , Ct +v, f 3 ghv 8 . C +v, t +v,= s +v, . y t +v, . 7t +v, @ Cv q Cv Cv
r
ru htxlydohqwo|
6 5 ]r f Ct +v, Ct +v, Ct +h v > v3 , 8 s +v, . y t +v, gh v . 7t +v, Cv Cv q Cv C +v, C +v, t +v, @ t +v, Cv Cv
r
+4:,
Zh qrz suryh wkh iroorzlqj uhvxow= Ohppd 5 Wkh rswlpdo txdqwlw| vfkhgxoh vdwlv hv wkh iroorzlqj frqglwlrq Ct +v, Ct +v, @ Cv Cv q iru hdfk v dqg hdfk l= Wkh uhvxow lv wulyldo zkhq wkh rswlpdo txdqwlw| lv t +v, @ th +v ,1 Zkhq t +v, @ te +v ,> qrwlfh wkdw wkh iroorzlqj krogv= Ce t +v, Ch t +v, 4 C+v, @ Cv Cv Cv dqg Ch t +v, 4 C+v, Ce t +v, @ Cv Cv Cv E| wkh gh qlwlrq ri +v,> zh fdq hdvlo| fkhfn wkdw wkh uhvxow1
C+v, C+v, @ zklfk qdoo| suryhv Cv Cv
E| Ohppd 5/ zh fdq uh0zulwh htxdwlrq +55, dv iroorzv ; < ] r Ct +h v > v3 , A A A gh v A A A @ C +v, Cv ? sf +v, . y t +v, r @ C +v, = 4. . A qA t +v, t +v, Cv Cv A A A A = > 56
uhsodflqj sf +v, e| lwv ydoxh/ dv gh qhg e| htxdwlrq +47,/ |lhogv ; < ] r ] r Ct +h v > v3 , A A A t +h v > v3 ,gh v gh v A A A @ C +v, Cv ? 6 r r @ C +v, = . . . 2 A qA 5 q ^t +v,` t +v, Cv Cv A A A A = > e| xvlqj djdlq Ohppd 5 wr uhsodfh zh jhw
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r
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t +h v > v3 ,gh v t +v,
zkhuh *+, lv d wzlfh gl huhqwldeoh ixqfwlrq gh qhg rq wkh vhw ^qv> qv` @ q = Vxpplqj xs/ vr idu zh kdyh vkrzq wkdw wkh rswlpdo phfkdqlvp pd| eh lpsohphqwhg e| d xqlirup sulfh vfkhgxoh li dqg rqo| li
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.
v 5q
zkhuh * lv d wzlfh gl huhqwldeoh ixqfwlrq vdwlvi|lqj wkh iroorzlqj lqwhjudo htxdwlrq ] r 5 6 t +h v > v3 ,gh v ] 9 : 9*+v . === . v? , . r . v : t +v > v3 ,i3 +v3 ,gv3 @ 3= 7 q t +v, 5q 8 l 3
ru dowhuqdwlyho|
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57
+4 ==> v3 > vn > ==> v? , m ' v @ y3 A y3 +v ,j zkhuh y3 +v , lv gh qhg dv wkh lqyhuvh ri vf +y3 ,> l1h1 f y3 +v , @
v +5 4,v 4
Dovr/ e| xvlqj wkh gh qlwlrq ri wkh rswlpdo txdqwlw|/ wkh OKV whup ri htxdwlrq +4 y3 ,i3 +v3 ,gv3 =
3
Qrz/ e| dsso|lqj wkh Jhqhudol}hg Fkdqjh Yduldeoh Wkhruhp +JYFW, zh fdq vhw y3 @ f 56 +v3 , iru hdfk l> zklfk qdoo| lpsolhv wkdw + f3 , @ ^y3 +v ,> +q 4,v` 5 U=
D pdlq lpsolfdwlrq ri wkh JYFW vd|v wkdw/ wkhuh h{lvwv d phdvxuh gh qhg ryhu f ^y3 +v ,> +q 4,v` vxfk wkdw ] ] *+v . y3 ,t +v > y3 ,i3 +v3 ,gv3 @
E
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3
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f Er
lf 3
3
E| xvlqj wkh uhvxow deryh/ wkh lqwhjudo htxdwlrq +4 y3 , +y3 ,gy3 @ j+v ,
f Er 3
55
D Yrowhuud lqwhjudo htxdwlrq ri wkh wkh uvw nlqg lv gh qhg lq wkh iroorzlqj zd|= ] E% s E%c + E%c + _+ ' }E% +3
lq rwkhu zrugv/ rqh ri wkh lqwhjudo h{wuhph pxvw ghshqg rq wkh yduldeoh % 56 Vhh Gxqirug dqg Vfkzduw} +4
